3. The SDDS EWMA-AV Chart
3.2. Detection Performance of the SDDS EWMA-AV Chart
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3.2. Detection Performance of the SDDS EWMA-AV Chart
To measure the performance of the SDDS EWMA-AV chart, we calculate the average run length (ARL) using the computer simulation of the process with 10,000 times. The of the SDDS EWMA-AV chart depends on the values of , , ,
, , , , , and . The choice of the design parameters has a high flexibility for a acceptably fixed , and we use the same direct search approach on the Section 2.2 to find the combination of the parameters ( , , ,
, , ) given , , , and by simulation.
Table 27. Parameters of the SDDS EWMA-AV chart with and for various , , and .
No.
E(n)
1 0.1 4 6 5 3.21 2.18 1.93 1.31 3.05 2.07 4.82 368.54 2 0.1 6 12 8 3.10 2.35 1.74 1.32 2.86 2.17 7.76 370.13 3 0.1 8 16 10 3.10 2.44 1.89 1.49 2.59 2.04 9.92 374.54 4 0.2 4 6 5 3.01 2.47 1.81 1.48 2.65 2.17 4.77 370.04 5 0.2 6 12 8 2.96 2.56 1.66 1.43 2.64 2.29 7.87 368.38 6 0.2 8 16 10 2.95 2.57 1.80 1.57 2.49 2.17 9.86 370.65 7 0.3 4 6 5 2.89 2.64 1.73 1.58 2.47 2.25 4.77 368.54 8 0.3 6 12 8 2.89 2.67 1.62 1.50 2.50 2.31 7.74 372.82 9 0.3 8 16 10 2.88 2.70 1.76 1.65 2.36 2.21 9.80 372.68 10 0.4 4 6 5 2.80 2.68 1.68 1.61 2.49 2.39 4.81 371.28 11 0.4 6 12 8 2.80 2.70 1.57 1.51 2.54 2.45 7.81 370.17 12 0.4 8 16 10 2.80 2.72 1.71 1.66 2.37 2.31 9.88 368.29
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Table 27 shows the coefficients of the control limits of the SDDS EWMA-AV chart for the combinations of the parameters with and . Here we consider 0.1–0.4, ( , , )=(4, 6, 5), (6, 12, 8), (8, 16, 10). It can be seen that the when increases, , and decrease and , and
increase.
The value of depends on the in-control distribution of data. We calculate the values of for the N(0,1), DE(0,1), , Unif , and Exp(1) distributions. Table 28 shows the values of and the coefficients of the control limits of the SDDS EWMA-AV chart for their corresponding distributions with and . Here, we consider ( , , ) (4, 6, 5), (8, 16, 10) for N(0,1) and ( , , ) (8, 16, 10) for DE(0,1), , Unif , and Exp(1). Table 29 shows the values of and the coefficients of the control limits of the SDDS EWMA-AV chart for their corresponding distributions with and . Here, we consider ( , , ) (4, 6, 5) for N(0,1), DE(0,1) and DE(0,1 ).
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Table 28. Parameters of the SDDS EWMA-AV chart with for various distributions, , and given .
distribution E(n)
N(0,1) 0.3173
4 6 5 2.87 2.63 1.72 1.58 2.52 2.31 4.79 372.45
8 16 10 2.84 2.63 1.76 1.63 2.49 2.30 9.87 370.65
DE(0,1/ ) 0.2706 8 16 10 2.87 2.63 1.78 1.63 2.43 2.23 9.85 372.50
0.2094 8 16 10 2.92 2.61 1.81 1.62 2.43 2.17 9.77 372.91
Unif 0.3466 8 16 10 2.83 2.64 1.75 1.64 2.43 2.27 9.72 369.57
0.2621 8 16 10 2.87 2.64 1.78 1.64 2.41 2.21 9.85 368.90
Exp(1) 0.2432 8 16 10 2.90 2.62 1.80 1.62 2.44 2.20 9.74 371.01
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Table 29. Parameters of the SDDS EWMA-AV chart with for various distributions given , , and .
distribution E(n)
N(0,1) 0.3173 4 6 5 2.98 2.74 1.79 1.64 2.65 2.44 4.70 504.90 DE(0,1)
DE(0,1 )
0.2706 4 6 5 3.06 2.64 1.84 1.58 2.74 2.36 4.70 503.01
For the out-of-control process, it is assumed that the variance has shifted and . A small out-of-control ARL indicates superior out-of-control detection performance of the control chart. Tables 30–33 show the out-of-control ARL and E(n) of SDDS EWMA-AV chart with and under 0.1–0.9, three combinations of and , and 0.1,0.4(0.1), respectively.
It can be seen that the out-of-control ARL decrease when is far away from
, the out-of-control E(n) increase for small or moderate shift and decrease for large shift. If and increase (in-control E(n) increases), then the out-of-control ARL decrease.
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3.3. Performance Comparison with Existing Control Charts
We compare the performance of the SDDS EWMA-AV chart with other existing variance control charts under different distributions of quality variables. The average time to signal (ATS) and the adjusted ATS (AATS) are considered to measure the performance of a control chart if the variable sampling intervals are adopted. For performance comparison, or values of the competing charts are fixed at (or very close to) an acceptable value, such as 370 or 500, with a fixed sample size,
, and then compare their out-of-control ARL or AATS.
If the in-control distribution of a quality variables is unknown, we consider different values of , which depends on the distribution, for comparison. Tables 34–35 show the performance comparison of the SDDS EWMA-AV chart, the EWMA-AV chart (Yang and Arnold (2014)) and the New EWMA chart (Yang and Arnold (2015)) for 0.1–0.9, with , and under 0.1 and 0.4, respectively. Table 36 shows the performance comparison of the SDDS EWMA-AV chart and the New EWMA chart for 0.1–0.9 with and under 0.3. It can be seen that the SDDS EWMA-AV chart has superior out-of-control detection performance than the EWMA-AV chart for small to large shifts in variance, 0.1–0.9, when , 0.4, and superior than the New EWMA chart for small to large shifts in variance, 0.1–0.9, when
, 0.3, 0.4.
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Table 34. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV chart and the New EWMA chart with and under
Table 35. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV chart and the New EWMA chart with and under
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Table 36. Performance comparison of the SDDS EWMA-AV chart and the New EWMA chart with and under 0.3. corresponding under various distributions.
Under a standard normal distribution, Table 38 shows the detection performance of the SDDS EWMA-AV chart, the Shewhart R, the Shewhart S, the VSI S, the DS S, and the DSVSI S charts with , and (Lee et al.
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, Table 41 shows the detection performance of the SDDS EWMA-AV chart, the SL charts (Mukherjee and Chakraborti (2012)) and the SC charts (Chowdhury et al. (2014)) with , and .
It can be seen that under the standard normal distribution, (1) the SDDS EWMA-AV chart has superior out-of-control detection performance than the Shewhart R chart for , superior than the Shewhart S chart for , inferior than the VSI S, the DS S and the DSVSI S charts for (see Table 38); (2) the SDDS EWMA-AV chart has superior out-of-control detection performance than the NLE, the CEW and the EWMA-AV charts for , superior than the Zhang’s IRC chart for , superior than the Zhang’s ISC chart for (see Table 39); (3) the SDDS EWMA-AV chart has superior out-of-control detection performance than the NP-M charts for , superior than the Khoo and Lim’s IRC chart for (see Table 40); (4) the SDDS EWMA-AV chart has superior out-of-control detection performance than the SL and the SC charts for 1 (see Table 41).
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Table 37. The values of and corresponding under various distributions.
N(0, ) DE(0, )
DE(0, ) Unif Exp( )
1.0 0.3173 0.2706 0.2094 0.3466 0.2621 0.2432
1.1 0.3625 0.3102 0.2447 0.3951 0.2985 0.2761
1.2 0.4043 0.3476 0.2797 0.4363 0.3326 0.3077
1.3 0.4424 0.3801 0.3116 0.4710 0.3642 0.3374
1.4 0.4739 0.4100 0.3443 0.5008 0.3941 0.3635
1.5 0.5042 0.4395 0.3739 0.5297 0.4208 0.3894
1.6 0.5325 0.4661 0.4011 0.5549 0.4460 0.4139
1.7 0.5561 0.4898 0.4268 0.5780 0.4698 0.4339
1.8 0.5789 0.5115 0.4518 0.5989 0.4909 0.4562
1.9 0.5981 0.5327 0.4738 0.6159 0.5112 0.4759
2.0 0.6180 0.5514 0.4941 0.6343 0.5303 0.4927
2.2 0.6502 0.5865 0.5319 0.6642 0.5651 0.5254
2.4 0.6770 0.6153 0.5660 0.6893 0.5949 0.5556
2.6 0.7004 0.6411 0.5944 0.7101 0.6203 0.5812
2.8 0.7212 0.6637 0.6202 0.7297 0.6423 0.6023
3.0 0.7394 0.6844 0.6423 0.7468 0.6637 0.6241
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Table 38. Performance comparison of the SDDS EWMA-AV chart, the Shewhart R, the Shewhart S, the VSI S, the DS S and the DSVSI S charts with ,
Table 39. Performance comparison of the SDDS EWMA-AV chart, the NLE, the CEW, the EWMA-AV, IRC and ISC charts with , and
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Table 40. Performance comparison of the SDDS EWMA-AV chart, the NP-M charts and the Khoo and Lim’s IRC chart with , and
Table 41. Performance comparison of the SDDS EWMA-AV chart, the SL charts and the SC charts with , and under N(0, ).
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The double exponential distribution DE(0, ) with in-control mean 0 and variance 1 and DE(0, 1) with in-control mean 0 and variance 2 are adopted for detection performance comparison of control charts. Table 42 shows the detection performance of the SDDS EWMA-AV chart, the EWMA-AV, the IRC (Khoo and Lim (2005)) and the NP-M charts (Ghute 2014) with , and under DE(0, ), Table 43 shows the detection performance of the SDDS EWMA-AV chart and the SL charts (Mukherjee and Chakraborti (2012)) with , and under DE(0, ), Table 44 shows the detection performance of the SDDS EWMA-AV chart, the SL charts and the SC charts (Chowdhury et al. (2014)) with , and under DE(0, ).
It can be seen that under DE(0, ), the SDDS EWMA-AV chart has superior out-of-control detection performance than the EWMA-AV, the Khoo and Lim’s IRC, the NP-M charts for (see Table 42); the SDDS EWMA-AV chart has superior out-of-control detection performance than the SL charts for 1 (see Table 43); under DE(0, ), the SDDS EWMA-AV chart has superior out-of-control detection performance than the SL charts and the SC charts for 1 (see Table 44).
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Table 42. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV, the Khoo and Lim’s IRC and the NP-M charts with , and
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Table 43. Performance comparison of the SDDS EWMA-AV chart and the SL charts with , and under DE(0, ).
Table 44. Performance comparison of the SDDS EWMA-AV chart, the SL charts and the SC charts with , and under DE(0, ).
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For the student's t distribution, Table 45 shows the detection performance of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart (Zou and Tsung (2010)) with , and under . It can be seen that the SDDS EWMA-AV chart has superior out-of-control detection performance than the NLE, the CEW and the EWM charts for .
For the uniform distribution, Table 46 shows the detection performance of the SDDS EWMA-AV chart, the EWMA-AV, the Khoo and Lim’s IRC and the NP-M charts (Ghute (2014)) with , and under Unif . It can be seen that the SDDS EWMA-AV chart has superior out-of-control detection performance than the EWMA-AV and the NP-M charts for , superior than the Khoo and Lim’s IRC chart for .
For the chi-square distribution, Table 47 shows the detection performance of the SDDS EWMA-AV chart, the NLE chart, the CEW chart and the EWM chart (Zou and Tsung (2010)) with , and under . It can be seen that the SDDS EWMA-AV chart has superior out-of-control detection performance than the EWM chart for , superior than the CEW chart for , inferior than the NLE chart for .
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Table 45. Performance comparison of the SDDS EWMA-AV chart, the NLE chart, the CEW chart and the EWM chart with , and
under . SDDS EWMA-AV chart
NLE chart
CEW chart
EWM chart
E(n) ARL ARL ARL ARL
1.0 9.77 372.91 370.00 128.00 362.00
1.1 10.51 82.05 218.00 88.30 242.00
1.2 11.57 29.10 127.00 65.60 175.00
1.3 12.37 15.50 80.70 49.00 130.00
1.4 13.05 10.02 56.70 38.10 103.00
1.6 14.00 5.65 34.30 24.50 68.90
1.8 14.73 3.88 24.20 17.40 49.90
2.0 15.34 3.03 18.80 13.10 38.30
3.0 16.54 1.67 9.21 5.78 16.40
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Table 46. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV, the Khoo and Lim’s IRC and the NP-M charts with , and
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Table 47. Performance comparison of the SDDS EWMA-AV chart, the NLE chart, the CEW chart and the EWM chart with , and
under . SDDS EWMA-AV chart
NLE chart
CEW chart
EWM chart
E(n) ARL ARL ARL ARL
1.0 9.85 368.90 374.00 118.00 366.00
1.1 10.61 90.82 23.60 72.90 224.00
1.2 11.63 32.85 11.00 48.40 153.00
1.3 12.46 17.77 7.49 34.40 109.00
1.4 12.99 11.60 5.69 25.80 82.20
1.6 13.89 6.52 4.06 16.50 53.50
1.8 14.66 4.46 3.35 11.70 37.90
2.0 15.11 3.45 2.92 9.03 28.80
3.0 16.14 1.82 2.05 3.99 12.10
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Table 48. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV and the Khoo and Lim’s IRC charts with , and
For the exponential distribution, Table 48 shows the detection performance of the SDDS EWMA-AV chart, the EWMA-AV and the Khoo and Lim’s IRC charts with , and under Exp( ). It can be seen that the SDDS EWMA-AV chart has superior out-of-control detection performance than the EWMA-AV chart and the Khoo and Lim’s IRC chart for .
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3.4. Example
The same non-normal service times example in Table 24 is used to illustrate the applications of the SDDS EWMA-AV chart. For double sampling scheme, we can let (
–
), (–
) and .To construct the SDDS EWMA-AV chart, can be estimated by pooled sample variance , the statistics , and in stage 1, and ,
, , and are calculated (see Table 49). Then, is estimated by
. Here, we assume the in-control data are close to population.
In reality, to reduce the error of estimation, the and are expected to be estimated by large in-control data.
Thus, the SDDS EWMA-AV chart with and is constructed as follows:
Stage 1 chart:
, ,
, (32) ,
Stage 2 chart:
,
. (33) Table 50 and Figure 5 show all plotted statistics fell in the CR, so no need to take the second sample.
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Table 49. The values of statistics , , j=1, 2, , 5, and for t
he in-control service times.
Stage 1 Stage 2
t
1 11.590 0.005 0.076 0 5.056 53.976 2.691 1 1
2 86.348 45.888 1.921 1 408.694 1.217 12.450 1 2
3 77.862 3.100 199.400 1 30.890 3.050 13.676 1 2
4 19.769 32.240 11.568 1 17.701 11.520 0.572 0 1
5 18.470 43.524 0.120 1 1.566 5.780 13.056 0 1
6 31.883 10.215 36.551 1 167.262 1.656 27.454 1 2
7 24.853 29.261 1.674 0 32.321 0.396 48.118 2 2
8 24.995 92.344 32.886 2 15.180 38.720 43.059 2 4
9 12.782 81.026 12.500 1 2.060 1.730 0.029 0 1
10 31.465 12.852 0.092 0 66.010 1.901 10.534 1 1
11 5.621 22.178 0.396 0 3.699 5.511 3.050 0 0
12 8.513 0.115 0.320 0 24.012 7.960 6.301 0 0
13 44.712 0.819 0.819 0 31.047 114.005 4.381 2 2
14 12.633 0.157 60.061 1 11.810 3.406 0.014 0 1
15 39.958 218.196 13.364 1 6.125 4.500 66.701 1 2
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Table 50. The plotting statistics of the SDDS EWMA-AV chart with for t=1, 2, , 15.
t Stage 1
Detect result
1 0 0.570 -0.926 IC
2 1 0.592 -0.190 IC
3 1 0.612 0.223 IC
4 1 0.631 0.520 IC
5 1 0.650 0.757 IC
6 1 0.667 0.956 IC
7 0 0.634 0.457 IC
8 2 0.702 1.316 IC
9 1 0.717 1.453 IC
10 0 0.681 0.978 IC
11 0 0.647 0.553 IC
12 0 0.615 0.170 IC
13 0 0.584 -0.179 IC
14 1 0.605 0.054 IC
15 1 0.625 0.268 IC
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Figure 5. The SDDS EWMA-AV chart for t=1, 2, , 15.
For future 10 days (t=16, 17, , 25), the new automatic service system of the bank branch is used. The new service times – are measured from 4 counters at first and statistic is calculated, if is within the , then the second sample of service times – are measured from other 6 counters. Table 51 shows the statistics and it can be seen that we need to take the second sample of 4 days (t=22–25). The statistic of the new service times is significantly reduced in Figure 6 and and are out-of-control.
For comparing the out-of-control detection performance, we constructed the corresponding EWMA-AV chart, transformed Shewhart S chart and transformed EWMA-S chart by applying transformation. The EWMA-AV chart with
, and , the transformed Shewhart S chart with L=3 and , and the transformed EWMA-S chart with L=2.492, and
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are constructed as follows.
EWMA-AV , EWMA-AV , ,
, (34) TEWMA-S ,
L TEWMA-S .
The EWMA-AV chart, transformed Shewhart S chart and transformed EWMA-S chart show no out-of-control variance signal, respectively (see Figure 7-9). It can be seen that the SDDS EWMA-AV chart has superior out-of-control detection than the EWMA-AV chart, transformed Shewhart S chart and transformed EWMA-S chart, in this example.
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Figure 6. The SDDS EWMA-AV chart for t=1, 2, , 25.
Figure 7. The EWMA-AV chart.
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Figure 8. The transformed Shewhart S chart.
Figure 9. The transformed EWMA-S chart
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